Range minimum query indexes in higher dimensions

Pooya Davoodi; John Iacono; Gad M. Landau; Moshe Lewenstein

doi:10.1007/978-3-319-19929-0_13

Range minimum query indexes in higher dimensions

Pooya Davoodi
, John Iacono
, Gad M. Landau
, Moshe Lewenstein

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Range minimum queries (RMQs) are essential in many algorithmic procedures. The problem is to prepare a data structure on an array to allow for fast subsequent queries that find the minimum within a range in the array. We study the problem of designing indexing RMQ data structures which only require sub-linear space and access to the input array while querying. The RMQ problem in one-dimensional arrays is well understood with known indexing data structures achieving optimal space and query time. The two-dimensional indexing RMQ data structures have received the attention of researchers recently. There are also several solutions for the RMQ problem in higher dimensions. Yuan and Atallah [SODA’10] designed a brilliant data structure of size O(N) which supports RMQs in a multi-dimensional array of size N in constant time for a constant number of dimensions. In this paper we consider the problem of designing indexing data structures for RMQs in higher dimensions. We design a data structure of size O(N) bits that supports RMQs in constant time for a constant number of dimensions. We also show how to obtain trade-offs between the space of indexing data structures and their query time.

Original language	English
Title of host publication	Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings
Editors	Ugo Vaccaro, Ely Porat, Ferdinando Cicalese
Publisher	Springer Verlag
Pages	149-159
Number of pages	11
ISBN (Print)	9783319199283
DOIs	https://doi.org/10.1007/978-3-319-19929-0_13
State	Published - 2015
Event	26th Annual Symposium on Combinatorial Pattern Matching, CPM 2015 - Ischia Island, Italy Duration: 29 Jun 2015 → 1 Jul 2015

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9133
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	26th Annual Symposium on Combinatorial Pattern Matching, CPM 2015
Country/Territory	Italy
City	Ischia Island
Period	29/06/15 → 1/07/15

Bibliographical note

Publisher Copyright:
© Springer International Publishing Switzerland 2015.

Funding

M. Lewenstein—Research supported by BSF grant 2010437, a Google Research Award and GIF grant 1147/2011. J. Iacono—Research supported by NSF grant CCF-1018370 and BSF grant 2010437. G.M. Landau—Research partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 571/14, Grant No. 2008217 from the United States- Israel Binational Science Foundation (BSF) and DFG.

Funders	Funder number
German-Israeli Foundation for Scientific Research and Development
Google Research Award
Israel Science Foundation
National Science Foundation
United States-Israel Binational Science Foundation
National Science Foundation	CCF-1018370, 0904246
United States - Israel Binational Science Foundation
Deutsche Forschungsgemeinschaft
German-Israeli Foundation for Scientific Research and Development	1147/2011
United States-Israel Binational Science Foundation	2010437
Israel Science Foundation	2008217, 571/14

Access to Document

10.1007/978-3-319-19929-0_13

Cite this

Davoodi, P., Iacono, J., Landau, G. M., & Lewenstein, M. (2015). Range minimum query indexes in higher dimensions. In U. Vaccaro, E. Porat, & F. Cicalese (Eds.), Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings (pp. 149-159). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9133). Springer Verlag. https://doi.org/10.1007/978-3-319-19929-0_13

Davoodi, Pooya ; Iacono, John ; Landau, Gad M. et al. / Range minimum query indexes in higher dimensions. Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings. editor / Ugo Vaccaro ; Ely Porat ; Ferdinando Cicalese. Springer Verlag, 2015. pp. 149-159 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Range minimum query indexes in higher dimensions",

abstract = "Range minimum queries (RMQs) are essential in many algorithmic procedures. The problem is to prepare a data structure on an array to allow for fast subsequent queries that find the minimum within a range in the array. We study the problem of designing indexing RMQ data structures which only require sub-linear space and access to the input array while querying. The RMQ problem in one-dimensional arrays is well understood with known indexing data structures achieving optimal space and query time. The two-dimensional indexing RMQ data structures have received the attention of researchers recently. There are also several solutions for the RMQ problem in higher dimensions. Yuan and Atallah [SODA{\textquoteright}10] designed a brilliant data structure of size O(N) which supports RMQs in a multi-dimensional array of size N in constant time for a constant number of dimensions. In this paper we consider the problem of designing indexing data structures for RMQs in higher dimensions. We design a data structure of size O(N) bits that supports RMQs in constant time for a constant number of dimensions. We also show how to obtain trade-offs between the space of indexing data structures and their query time.",

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Davoodi, P, Iacono, J, Landau, GM & Lewenstein, M 2015, Range minimum query indexes in higher dimensions. in U Vaccaro, E Porat & F Cicalese (eds), Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9133, Springer Verlag, pp. 149-159, 26th Annual Symposium on Combinatorial Pattern Matching, CPM 2015, Ischia Island, Italy, 29/06/15. https://doi.org/10.1007/978-3-319-19929-0_13

Range minimum query indexes in higher dimensions. / Davoodi, Pooya; Iacono, John; Landau, Gad M. et al.
Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings. ed. / Ugo Vaccaro; Ely Porat; Ferdinando Cicalese. Springer Verlag, 2015. p. 149-159 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9133).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Davoodi P, Iacono J, Landau GM, Lewenstein M. Range minimum query indexes in higher dimensions. In Vaccaro U, Porat E, Cicalese F, editors, Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Proceedings. Springer Verlag. 2015. p. 149-159. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-19929-0_13

Range minimum query indexes in higher dimensions

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